Surface defects on E-string from 5-brane webs

We study 6d E-string theory with defects on a circle. Our basic strategy is to apply the geometric transition to the supersymmetric gauge theories. First, we calculate the partition functions of the 5d SU(3)$_0$ gauge theory with 10 flavors, which is UV-dual to the 5d Sp(2) gauge theory with 10 flavors, based on two different 5-brane web diagrams, and check that two partition functions agree with each other. Then, by utilizing the geometric transition, we find the surface defect partition function for E-string on $\mathbb{R}^4\times T^2$. We also discuss that our result is consistent with the elliptic genus. Based on the result, we show how the global symmetry is broken by the defects, and discuss that the breaking pattern depends on where/how we insert the defects.


Introduction
Study of surface defects [1,2] provides us with a tool for uncovering novel structure and enriching our understanding of non-perturbative aspects and interplay between theories of codimension 2 [3][4][5][6][7][8][9].A way to see how the defects affect physical systems is to calculate the partition function.It is known that partition functions for some class of 5d N = 1 supersymmetric gauge theories compactified on a circle is equivalent to the topological string amplitudes on corresponding non-compact toric Calabi-Yau manifolds under suitable parameter correspondence, which is known as geometric engineering [10][11][12][13].Based on the equivalence between toric Calabi-Yau manifold and 5-brane web [14], we can translate the geometry into brane set up and vice versa.In the presence of defects, the partition function can be computed by implementing 5d system with defects on a Type IIB 5-brane web, where the defects are realized as (perpendicular) D3-branes inserted on the (p, q)-web plane for supersymmetric gauge theories.Such defect insertion is captured as a particular choice of Kähler parameters in topological string amplitudes, which corresponds to open topological string amplitude [15,16], which is known as geometric transitions [17,18].
The procedure of obtaining defect partition functions can be viewed as a generalization of Higgsing procedure from 5-brane webs.Depending how to choose or tune the Kähler parameters, one sees usual Higgsing or a system with defect.More precisely, to obtain defect partition function, we need to tune Kähler parameters in a way that not only it reduces the rank of gauge group but also yields the open topological string partition function up to Coulomb branch independent overall factors such as MacMahon function or the extra factors.Geometric transition when the number of inserted D3 defects is zero, reduces to usual Higgsing.In this regard, it is a generalization of Higgsing procedure, and we refer to it as defect Higgsings.Using the defect Higgsing, the 5d defect partition functions have been computed in a straightforward manner.
Many 6d theories on a circle are also realized on 5-brane webs, one can hence also apply defect Higgsing to such KK theories.One of example is M-string theory with surface operators inserted [19], where M-string is realized as a periodic (p, q) 5-brane webs where the NS5-branes are identified along the a circle of radius R −1 which corresponds to the compactification circle radius of 6d theory on a circle [20,21].
As another example of defect Higgsing on KK theories, in this paper, we study 6d E-string theory with surface operator.E-string theory on a circle of radius R is realized as SU(2) gauge theory with 8 hypermultiplets in the fundamental representation (flavors).There are two different 5-brane webs for the SU(2) gauge theory.One is of spirally periodic shape whose period is identified as R −1 , which is called Tao diagram [22].The other is of two O5-planes with their distance R −1 .Since an orientifold plane is used to realize Sp or SO gauge group in general, we refer to the former 5-brane web as web diagram for SU (2) gauge theory, while the latter as web diagram for Sp (1) gauge theory in this paper even though SU (2) and Sp(1) are identical.Partition function can be computed based on 5-brane webs via topological vertex [23].In particular, topological vertex formalism in the presence of O5-planes is also developed in [24] and both topological string partition functions agree with the elliptic genus of 6d E-string theory.
Our strategy of computing E-string partition function with defects is to first consider 5d SU(3) 0 gauge theory with 10 flavors and then apply the defect Higgsing.The 5d SU(3) 0 gauge theory with 10 flavors is UV-dual to the 5d Sp(2) theory with 10 flavors, in the sense that their UV completion is the same [25,26].The corresponding 6d theory description is given by 6d Sp(1) gauge theory with 10 flavors and a tensor multiplet.Together with the brane configurations, these dual descriptions allow one to compute the partition function in several different ways.For instance, the ADHM-like method [27], the elliptic genus [28], and the topological vertex method [29].We compute the defect partition function by using topological vertex method and compare it with the elliptic genus for E-string with defects which is obtained from 6d Sp(1) gauge theory with 10 flavors.
Analogous to the case with the 5d SU(2) gauge theory, the 5d SU(3) 0 gauge theory with 10 flavors has two different 5-brane configurations: One is without O5-plane and the other is with two O5-planes.The 5-brane configuration without O5-plane is Tao diagram [22].This Tao diagram can be obtained from the 5-brane web diagram with two O7 −planes, which is T-dual of the type IIA brane setup with an O8 − -plane [30,31] for 6d Sp(1) gauge theory with 10 flavors and a tensor multiplet.By resolving both O7 − -planes into two different 7-branes [32], respectively, we obtain the diagram for the 5d SU(3) 0 gauge theory, which can be deformed to be a Tao diagram.On the other hand, if we resolve only one O7 − -plane, we obtain the diagram for the 5d Sp(2) gauge theory, which explains the UV-duality between the SU(3) gauge theory and the Sp(2) gauge theory [25,26,29].The 5-brane configuration with two O5-planes is again the T-dual of the type IIA brane setup but with O6-plane instead of O8-plane.As an abuse of notation, we refer to the former web diagram without O5-planes as the web diagram for SU(3) gauge theory, while the latter diagram with two O5-planes as the web diagram for Sp (2) gauge theory in this paper, based on the knowledge that orientifold planes are used to realize Sp or SO gauge group in general. 1he organization of the paper is as follows.In section 2, we review the partition function for 5d SU(3) 0 gauge theory with 10 flavors from two different 5-branes setups: one with a 5-brane web without O5-planes and the other with two O5-planes.In section 3, using 5d 5-brane configurations for the SU(3) 0 gauge theory with 10 flavors, we perform defect Higgsing to yields 5d SU(2) gauge theory with 8 flavors.For comparison, we implement the defect Higgsing to the elliptic genus partition function of E-string theory to the agreement.In section 4, we discuss issue of global symmetry in the presence of defects.We then conclude and discuss unbroken global symmetry and possible generalizations.In Appendix, we discuss our conventions, decoupling limit to get 5d theories from KK theories, and defects on pure SU (2) theories with different discrete theta angles.

5d SU(3) gauge theories, 5-brane webs, and UV duality
In this section, we consider the 5d N = 1 SU(3) κ gauge theories from the perspective of 5-brane webs in Type IIB string theory.In particular, we discuss how to obtain the BPS partition function of the SU(3) 0 gauge theory with 10 flavors of the Chern-Simons level κ = 0.The computation is performed based on two different 5-brane web diagrams.One is a 5-brane web diagram without O5-planes, which is Tao diagram introduced in [22].Though it is spirally periodic, one can apply the topological vertex method to compute unrefined Nekrasov partition function.The other is a 5-brane web with two O5-planes.It is also possible to implement the topological vertex method to the 5-brane configurations with O5-plane(s) with special deformation of web diagrams and careful Young diagram assignments near O5-planes [24].
The partition function for 5d SU(2) with 8 flavors has been already computed explicitly [22,24,33].The resulting partition functions based on a Tao diagram [22] and a 5-brane with two O5-planes [24] look quite different, however, they agree up to unphysical factors, called extra factor, which do not depend on the Coulomb branch parameters.As for the 5d N = 1 SU(3) 0 gauge theory with 10 flavors, the topological string partition function based on a Tao diagram has been already computed [29], but it has not been computed explicitly based on a 5-brane web with two O5-planes yet.In section 2.1, we first review the computation with the topological vertex based on the Tao diagram.Then, in section 2.2, we compute the partition function based on the 5-brane web with two O5-planes.We refer to the former partition function as the partition function for 5d N = 1 SU(3) 0 gauge theory with 10 flavors, while the latter one as the partition function for 5d N = 1 Sp(2) gauge theory with 10 flavors.We will see agreement between them by expressing the latter partition function in terms of the parameters of the SU(3) gauge theory.The detailed computations and some notations are summarized in Appendix A.

Partition function from Tao diagram
We first briefly review the computation of the partition function for 5d N = 1 SU(3) 0 gauge theory with 10 flavors based on a 5-brane web [29].A 5-brane configuration for such 5d marginal theories is called Tao web diagram [22], which is of a spiral sharp with a periodic structure whose periodicity is given by the instanton factor squared.A Tao web diagram for 5d N = 1 SU(3) 0 gauge theory with 10 flavors is depicted in Figure 1(a).Though this 5-brane configuration, in principle, has infinitely many Kähler parameters, they are not independent due to the periodic structure of the web diagram, and only 13 parameters corresponding to the two Coulomb moduli parameters, ten mass parameters, and one instanton factor are independent.For computations, only half of the diagram suffices due to symmetry of the web diagram.For instance, in order to compute Figure 1(a), we only need to consider Figure 1(b).This diagram in Figure 1(b) can be further untangled with basic building blocks painted in different colors: middle strips, spiral strip 1, and spiral strip 2 as in Figure 2. The assignment of the independent Kähler parameters (Q i ) and Young diagrams (µ I ) are depicted in Figure 2. The (unrefined) partition function is then obtained by gluing these building blocks with suitable edge factors.
The partition function for the 5d SU(3) 0 gauge theory with 10 flavors then reads Here, Z glue are the three edge factors for gluing, given by where q = e −β is the parameter associated with the string coupling, defined with the selfdual Ω-deformation parameter = 1 = − 2 .Z half 1 and Z half 2 are the contributions of  H z e M e u 2 w n O K q Z l 2 U Z Y c r q k G z 7 m q q / G i Z X N J l z V e k G u 7 v f t C g 9 u O a h q H b t P i Z V 2 q G u q J q k g u U d l M R a x E Y i z J v I g O A t E H M f i x b 0 Y e c I R j m F B Q h w 4 O A y 5 h D R I c G i W I Y L C I K 6 N F n E 1 I 9 e 4 5 2 g i R t k 5 Z n D I k Y m u 0 V u l U 8 l m D z r 2 a j q d W 6 B W N p k 3 K K O L s m d 2 x L n t i 9 + y V f f 5 Z q + X V 6 H l p 0 i 7 3 t d y q h M 8 X s x / / q n T a X Z x + q 4 Z 6 d n G C L c + r S t 4 t j

Q2
< l a t e x i t s h a 1 _ b a s e 6 4 = " f W h j b w j j  x g h T S 5 E X H 5 V p V N / m W f r j q 3 2 / V u e s J 2 9 q Q R w 7 f r W r 7 l t g T h i a J 2 t 6 R w q z w W L a s l q N x l m J B < l a t e x i t s h a 1 _ b a s e 6 4 = " + r T H S D l 7 i t k g a 6 9 < l a t e x i t s h a 1 _ b a s e 6 4 = " G 1 n n f 7 W t Y m T N n 7 r l z h q u 7 p v A D x p 6 6 p O 6 e 3 r 7 + 1 M D g 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " q Z 4 9 j u k o y e 7 j v X f e e f f c d x 5 X M l T F s h n r e Y S J S e / U t G / G P z s 3 v x A I L i 7 l L b 1 p y j w n 6 6 p u F i X R 4 q q i 8 Z y t o y e 7 j v X f e e f f c d x 5 X M l T F s h n r e Y S J S e / U t G / G P z s 3 v x A I L i 7 l L b 1 p y j w n 6 6 p u F i X R 4 q q i 8 Z y t o y e 7 j v X f e e f f c d x 5 X M l T F s h n r e Y S J S e / U t G / G P z s 3 v x A I L i 7 l L b 1 p y j w n 6 6 p u F i X R 4 q q i 8 Z y t o y e 7 j v X f e e f f c d x 5 X M l T F s h n r e Y S J S e / U t G / G P z s 3 v x A I L i 7 l L b 1 p y j w n 6 6 p u F i X R 4 q q i 8 Z y t o y e 7 j v X f e e f f c d x 5 X M l T F s h n r e Y S J S e / U t G / G P z s 3 v x A I L i 7 l L b 1 p y j w n 6 6 p u F i X R 4 q q i 8 Z y t o y e 7 j v X f e e f f c d x 5 X M l T F s h n r e Y S J S e / U t G / G P z s 3 v x A I L i 7 l L b 1 p y j w n 6 6 p u F i X R 4 q q i 8 Z y t < l a t e x i t s h a 1 _ b a s e 6 4 = " w l h y m Z e c 1 a d 2 e 7 Y l D U B O I 3 Y m T N n 7 r l z h q s 5 h v B 8 x h 7 b p P a O z q 7 u W E + 8 t 6 9 / I J E c H F r 2 7 L q r 8 4 p u G 7 a 7 q q k e N 4 T F K 7 7 w D b 7 Y m T N n 7 r l z h i u Z q m I 7 j L X 7 h P 6 B w a H h w M j o W H B 8 Y j I 0 N Z 2 1 j Z o l 8 4 x s q I a V l 0 S b q 4 r O M 4 7 i q D x v W l z U J J X n p O q m e 5 + r c 8 t W D H 3 P a Z i 8 q r a 0 W l z b X 5 1 0 p 9 i V + y F / F + y J 3 Z P P 7 B q r 9 p 1 n q + e / e F H J S 9 B e + T v z f g J i p m 0 n E 1 n 8 L w e 0 q w 0 t t y v 9 x 8 P 5 9 3 9 V B u 0 e 9 r 5 U f 3 r 2 U M V s 6 F U n 7 3 b I B L / Q G v r 6 4 e l r f n 5 1 3 J 9 g V + y F / F + y J 3 Z P P z D r b 9 p 1 j q + e / e F H J S 9 B e + S f z L w e 0 q w 0 t t y v 9 x 8 P 5 9 3 9 V B u 0 e 9 r 5 U f 3 r 2 U M V s 6 F U n 7 3 b I B L / Q G v r 6 4 e l r f n 5 1 3 J 9 g V + y F / F + y J 3 Z P P z D r b 9 p 1 j q + e / e F H J S 9 B e + S f z

3
< l a t e x i t s h a 1 _ b a s e 6 4 = " y p 4 q t t Q p l U P y   the half diagram in Figure 1(b) and the other half, which are composed of three building blocks in Figure 2, where Each of which takes the form2

.5c)
Here we define some quantities: ν i are the Young diagrams along the horizontal lines.R µν (Q) is a function giving the contributions coming from the string wrapping on the internal line characterized by Kähler moduli Q with Young diagrams µ and ν, Z half2 can be obtained by suitable parameter replacement as the lower half (half1) and upper half (half2) of the diagram are symmetric.
After removing the extra factor, and using the proper flop transition (A.12) for the perturbative part, one finds that where {A} = {A 1 , A 2 , A 3 } and {M } = {M 1 , M 2 , ..., M 10 } are Coulomb branch moduli and mass parameters, respectively.We just provide the explicit map between Kähler parameters and physical parameters in Appendix B. A detailed derivation is given in [29].We also define F 0,1,2 (A, M i ) as ) where and χn = χ0 A letter "(cyclic)" means two more terms that are obtained by taking a cyclic permutation of A I (I = 1, 2, 3) on the first term.{A 2 } and {M 2 } denote the squares of A I and M i , (2.11)

5-brane web with two O5-planes
We now compute the partition function based on a 5-brane web with two O5-planes.The corresponding web describes 5d Sp(2) gauge theory with 10 flavors.As it is UV-dual to 6d Sp(1) gauge theory with 10 flavors and a tensor multiplet, a brane configuration for 5d Sp(2) gauge theory with 10 flavors can be obtained as a T-dual version of a Type IIA brane configuration for the 6d theory, which is made out of D6-branes, an O6-plane, and NS5 x 0 x M Y o 6 e 6 I 5 e 6 J H u q U P v v 9 Z q B T V 8 L 0 2 e 1 a 5 W 2 O X 4 6 V T + 7 V 9 V j W c P l U / V n 5 4 9 H G I l 8 K q z d z t g / F t o X X 3 j + P w l v 5 a b a 8 3 T N T 2 z / y t q 0 w P f w G y 8 a j d b I n e J G H + A / P 2 5 f 4 m l M l 2 q h I J N e P 4 F s r 0 J 9 i T i r s = < / l a t e x i t > y 0

5
< l a t e x i t s h a 1 _ b a s e 6 4 = " < l a t e x i t s h a 1 _ b a s e 6 4 = " P C F x Z 6 0 7 u N q K g o 3   < l a t e x i t s h a 1 _ b a s e 6 4 = " g U < l a t e x i t s h a 1 _ b a s e 6 4 = " I P P t l M z q Z 1 y i d m < l a t e x i t s h a 1 _ b a s e 6 4 = " I P P t l M z q Z 1 y i d m The dashed lines and the solid lines denote the O5-planes and (p, q) 5-branes, respectively.The variables x I (I = 1, 2, 3) and y i (i = 1, 2, ..., 10) denote the position of the horizontal legs measured from the middle of the dashed line, and q denotes the instanton factor.
brane as given in Figure 3(a) [30,31].The corresponding Type IIB 5-brane configuration for 5d Sp(2) gauge theory with 10 flavors is a 5-brane configuration with two O5-planes as given by Figure 3(b).As discussed in [34], a 5-brane web diagram in Figure 3(b) can be deformed into a 5-brane web diagram given in Figure 4(a).Such deformed diagram gives rise to a strip-like diagram when the fundamental configuration is chosen to combine a half from the original diagram and the other from its reflected image due to the O5plane, as depicted in Figure 4(b), where the top of (1, 1)-brane is glueing to the bottom of (−1, −1)-brane by flipping the sign of the charges, i.e. the web diagram is compactified with the period q 2 .This is a crucial 5-configuration which enables one to implement the topological vertex method to a 5-brane web with O5-plane(s) [24].The computation in fact can be done in a straightforward way.With the fundamental configuration as a strip-like diagram given in Figure 4(b), one performs the topological vertex computation with the following additional procedures: (i) the framing factor associated with a 5-brane crossing an O5-plane needs to be shifted by 1. (ii) internal edges of the web diagram associated with color D5-branes are glued in the following way: as the configuration on the right hand side of Figure 4(b) are reflected, one takes the transpose for the Young diagrams assigned to the internal edges, and then one glues the these internal edges together.See Appendix C.1 for more details.
Based on the 5-brane configuration in Figure 4(b), the partition function for 5d Sp(2) gauge theory with 10 flavors is expressed in terms of a Young diagram sum over µ 1 , µ 2 , µ 3 , as where f µ is the framing factor defined in (A.5).The positions x I and y i correspond to the fugacities associated with the color D5-branes and those for 10 flavor masses, respectively.
For convenience, we have used Θ µν (Q) defined as We note that Θ µν (Q) can be expressed as the Jacobi theta functions (A.7) by using the analytic continuation formula [20], and hence it is of periodic structure.Notice that there are three Young diagram sums associated with color D5-branes in (2.12).As discussed earlier, the 5d N = 1 Sp(N ) gauge theory with N f flavors is dual to the 5d N = 1 SU(N + 1) κ gauge theory with N f flavors and the Chern-Simons level κ = N + 3 − N f /2, through some kind of geometric transition [25,29,35,36].This enables us to relate the parameters of the Sp(N ) theory and those of the SU(N + 1) theory.In this case, with a shifting factor q 1/2 Λ SU(3) for the Coulomb branch where , the parameter map between the Sp(2) gauge theory with 10 flavors (x I , y i ) and the SU(3) 0 gauge theory with 10 flavors (A I , M i ) is given by [29] x I = q 1/2 A I Λ SU(3) (I = 1, 2, 3), (2.14a) where the SU(3) Coulomb branch moduli A I satisfy A 1 A 2 A 3 = 1.We note that the instanton factors for these two dual theories are same q = q, as they are associated with the compactification radius.
With the SU(3) gauge theory parametrization, we find the partition function of SU(3) 0 gauge theory with 10 flavors in a symmetric form, where Z extra is an Coulomb branch moduli independent part which is given by where (a; q) ∞ = ∞ k=0 (1 − aq k ) is so-called Pochhammer symbol.There are in fact more Coulomb branch independent part in (2.15), as a whole we call them the extra factor.When obtaining the partition function, we mod out such extra factor from the topological string partition function.From here on, we neglect the extra factor.
By expressing the partition function (2.15) as an expansion of the instanton factor q, we can write the partition function for SU(3) 0 gauge theory with 10 flavors as the Plethystic exponential, it takes the following form where F n ({A}, {M }) are exactly the same as those obtained from Tao diagram (2.8a), (2.8b), and (2.8c) since they are the dual to each other, although the partition functions Z SU(3)+10F and Z Sp(2)+10F are derived from completely different diagrams 3 .Due to computational complication, we only presented terms of quadratic order in q, but can be checked the equivalence to higher orders.

Defect partition function of E-strings
In this section, we consider defect insertion to the E-string theory, from the 5-brane perspective.Through the geometric transitions, we can introduce a codimension-two defect which is a topological brane wrapping on the Lagrange submanifold in the topological string theory [15].We consider the partition function of the topological string in the presence of the topological brane, which we call, for short, the defect partition function.
To obtain the defect partition function 4 for the E-string theory, we utilize geometric transition [17].The procedure is to set the Kähler parameters associated with Higgsing Q to the M -th power of the exponential of the string coupling constant, Q = q M , where M is the number of the topological branes.In the context of the gauge theory, when M = 0, it becomes the usual Higgsing, as Q = 1 reduces the number of Coulomb branch moduli.When M > 0, it corresponds to the defects in the E-strings.We call such a procedure for introducing the defects a defect Higgsing.More concretely, when we consider the 5d N = 1 Sp(N ) gauge theory with 2N + 6 flavors for N ≥ 2, the defect Higgsing means In the following, we choose N = 2 but M > 0 arbitrary, which is relevant for M defects in the E-strings.We explicitly compute the defect partition function for the E-string based on 5-brane webs by implementing the defect Higgsings.

Defect Higgsing in web diagram
Before we consider the defect Higgsing, we first review the usual Higgsing procedure in 5-brane configurations with or without O5-planes.As done in the previous section, we consider 5d N = 1 SU(3) gauge theory with 10 flavors and then consider N = 1 Sp(2) gauge theory with 10 flavors.We will show that one obtains SU(2) gauge theory with 8 flavors, as a result of the Higgsing on both theories.
Usual Higgsings.As discussed earlier, 5-brane web for SU(3) gauge theory with 10 flavors is a Tao diagram given in Figure 1(a).To apply a Higgsing to the SU(2) theory with 8 flavors, we consider a strip consisting of a color D5-brane and two flavor branes that are connected.Unlike typical 5-brane web, for a Tao diagram, such strip is a spiral strip as painted in red in Figure 5(a).A usual Higgsing is then realized by assigning the relevant Kähler parameters to 1, so that such a spiral string can be Higgsed away, reducing the dimension of the Coulomb branch by one and the number of flavors by two.A diagrammatic procedure is depicted in Figure 5.
For implementing the Higgsing in the topological vertex calculation, as before, we can consider the half of the diagram in Figure 1(b).More specifically, in Figure 2(a), say Q2 is the Kähler parameter associated with the Higgsing, then we set Q2 = 1.We expect that the part involving the summation over the Young diagram µ 2 associated with the edges that are Higgsed is factorized in the partition function.In particular, the contribution involving the Young diagram µ 2 in the building block Z middle in (2.5a) becomes which vanishes unless µ 2 = ν 3 due to the factor R µ 2 ν 3 (1) 5 , so that (3.3) becomes 1 under the Higgsing.By combining another half building blocks, the summation of µ 2 is decoupled from other summations, as we expected, where Z SU(2)+8F is the partition function of SU(2) gauge theory with 8 flavors [22], with and Z Tao Framing can be interpreted as the contribution coming from the framing denoted by the red-colored strips in Figure 5, which is given by Now we consider the Higgsing from the 5-brane web in Figure 6 for the 5d N = 1 Sp(2) gauge theory with 10 flavors. 6The Higgsing procedure is almost the same as that on the Tao diagram.We align a color brane and two flavor branes, for instance, It is also straightforward to see that the partition function (2.12) vanishes unless the associated Young diagram is µ 2 = ø, and hence it yields that the partition function (2.12) reduces to Here Z Sp(1)+8F is the partition function for the 5d N = 1 Sp(1) gauge theory with 8 flavors, given by where 10}.The factor Z O5 Framing is an extra factor which comes from the contribution of the framing denoted by the red line in Figure 6, (1 − q i+j−1 ).(3.13) 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " w L s e Z 9 n Q q < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 9  < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 m i r J By rescaling the Coulomb moduli parameters and the mass parameters as, we see that (3.11) agrees with the partition function directly calculated from the web diagram of 5d N = 1 SU(2) with 8 flavors discussed in [24].
Defect Higgsings.Form here on, we consider a 5-brane system with defect D3-branes, which can be introduced in a similar way as for usual Higgsings.As discussed, the defect Higgsing is the Higgsing by setting the associated Kähler parameter not 1 but q M [4, 15,39].
Here, M is the number of defect D3-branes, and hence the defect Higgsing becomes a usual Higgsing when M = 0.For instance, in Figure 7(a), 5-brane web for the 5d Sp(2) theory with 10 flavors can be used to describe the 5d Sp(1) theory with 8 flavors and defects, where the 5-branes painted in red are defect Higgsed.The associated Kähler parameter is set to be q M , while the other Kähler parameter is just treated as a usual Higgsing.After having defect Higgsed, we denote the defect Higgsed part as a (blue) dotted line, representing M D3-branes.Likewise, such defect Higgsing can be performed in a Tao diagram in the same way, as depicted in Figure 7(b).
Implementing the defect Higgsings to the partition function is straightforward.For the partition functions that we obtained from topological vertex based on the web diagrams either with O5-planes or with the Tao diagram, in the previous section, we first set one of the Coulomb branch moduli A 2 to where we defined z = M 3 as a defect moduli.When M = 0, it reduces to (3.9), as expected.
As the computation is similar to the usual Higgsing, we summarize the results for both cases: For the partition function (3.5) obtained from the Tao web diagram, the defect Higgsing on SU(3) gauge theory with 10 flavors yields where and we have rescaled some variables as follows, to obtain the traceless condition of SU(2), A 1 A 3 = 1.This then naturally shifts other mass parameters For the partition function (2.15) obtained from the 5-brane web with O5-planes, the defect Higgsing on the Sp(2) gauge theory with 10 flavors yields where Z Sp(1)+8F build is defined in (3.12) with the replacement Λ SU(2) → q M/4 Λ SU(2) , and .
(3.21) (3.16) and (3.20) look different but they are the same when we expand as q and A 1 .
By expanding the partition function as a series of the instanton factor and the defect moduli, the partition function is decomposed into three parts; the physical part depending only on the Coulomb moduli F defect n (A), the Coulomb moduli and the defect moduli D defect n (z, A), up to an extra factor, where A = A 1 .We list the explicit forms of F defect n (A) and D defect n (z, A) up to 2-instanton, and and − q 13M/2 χ 0 χ 1 z 6 + q −13M/2 χ 7 χ 8 z −6 + q 5M χ 0 χ 2 z 9/2 − q −5M χ 6 χ 8 z −9/2 − q 7M/2 χ 0 χ 3 + q 5M/2 χ 0 χ 7 z 3 + q −7M/2 χ 5 χ 8 + q −5M/2 χ 1 χ 8 z −3 For the concise expression, we define U(8) characters, Under performing the defect Higgsing, it is interesting to check whether the global symmetry is broken.The usual Higgsing from the SU(3) gauge theory with 10 flavors to the SU(2) gauge theory with 8 flavors preserves an E 8 symmetry.Seeing the (enhanced) global symmetry from the instanton partition function is sometimes non-trivial as one needs to suitably combine flavors masses and instanton fugacity to form the characters of the enhanced global symmetry.As the SU(3) gauge theory with 10 flavors is a marginal theory, its partition function agrees with the corresponding elliptic genus partition function, which show global symmetry manifest.
For this reason, the global symmetry of the theory in the presence of the defects may not be manifest in (3.22), though they are expressed as the U(8) characters.To make it manifest, we consider the the elliptic genus.
• The duality map between 6d Sp(1) +10F + 1T and 5d SU(3) 0 + 10F, which comes from (3.31) and (2.14) with q = q: (3.32) Using the duality map above, one can readily check that two partition functions do agree with each other by a double expanding in terms of the 5d Coulomb parameter A 1 and the instanton fugacity q [29].Some comments are in order.First, this transformation is slightly different from the one given in [29], however, the difference is just the convention, and our convention is more useful to consider the flavor decoupling limit that we will discuss in Appendix D, so that we adopt them.Second, we check the agreement of the partition function and the elliptic genus by the double expansion as power series of A 1 and q up to second order through the duality map (3.32).
The usual Higgsing from 6d Sp(1) gauge theory with 10 flavors and a tensor, to 6d E-string theory, is achieved by setting the Coulomb branch parameter Ã and two mass parameters ỹ3 , ỹ8 as which is equivalent to (3.9) through the map (3.31).Then the elliptic genera of one and two strings become which agree with those for the E-string theory up to redefinitions of the parameters.For the elliptic genus of one string, the summation of the product of the theta functions can be expressed as E 8 theta function which is defined as the summation over the E 8 root lattice Γ 8 , where w ), and µ i = 1 2πi log y i , τ = 1 2πi log q.One can also express the elliptic genus of two strings as combinations of the E 8 theta functions, hence the E-strings enjoy E 8 Weyl symmetry.Now we implement the defect Higgsing.In a similar fashion, we set which is consistent with (3.15) with the duality map (3.32).The resulting elliptic genera of the E-strings with the defect are given by Zdefect where z = ỹ3 is the position of the defect insertion.We note here that unlike the usual Higgsing, ZDefect and ZDefect now are not invariant under E 8 Weyl symmetry, rather they are invariant under the SO(16) Weyl symmetry, .38)This implies that the E 8 global symmetry of the elliptic genus for the E-string is broken SO( 16) due to the presence of the defects.
To see the symmetry breaking more explicitly, let us consider the defect Higgsing of the elliptic genus expressed by the characters given in [38].A part of the contributions to the elliptic genus of one string Z(1) can be expressed as the character,

+ ZSU
(1) + O(q 3/2 ), (3.39)where we decompose Z(1) into two parts: the contributions involving SO(20) characters and those not depending on SO(20) characters, ZSO, SU = −χ where relevant the SO(20) characters are given by χ as well as the SU(2) character, By the defect Higgsing (3.36), the part of the elliptic genus of one string (3.40a) reduces to , (3.43) where we further introduce the SO(16) characters, have to vanish, however, there is no such solution except for trivial cases, M = 0. Therefore, we conclude that the symmetry of E-string is broken to SO( 16) by introducing the defect.

Global symmetry
In this section, we comment on global symmetry in the presence of the codimension-2 defects.In computing the defect partition function, we used geometric transitions, where we construct "unHiggsed" 5-brane configuration and perform the defect Higgsing.As unHiggsed theory does not have any defect attached, the defects are introduced through defect Higgsing.Theories with defects which can be geometrically engineered hence captures some subalgebra of global symmetry of the unHiggsed theory.For instance, the partition function for the E-string theory with defects that we have computed in the previous section show manifest SO( 16) global symmetry.We obtained it through the defect Higgsing of 6d Sp(1) gauge theory with 10 flavors which has SO (20) global symmetry.The defect of the E-string that we considered thus naturally has an SO (16) which can be understood as SO (20) global symmetry of the unHiggsed theory is broken to SO(16) along the defect Higgsing.
In M-theory perspective, E 8 global symmetry of E-string is the symmetry that arises as E-string probes M9 brane.Defects on E-string are expected not to disturb M9 brane, and hence one may expect to still see an E 8 symmetry for E-string theory with defects.Global symmetry of defect partition function, however, may depend on how we compute the defects.For instance, we can introduce different codimension-2 defects via geometric transitions for a given 5d theory by considering defect Higgsings on different unHiggsed 5-brane webs.These different defect Higgsings may capture different global symmetry, which could be the perturbative symmetry or more than the perturbative symmetry.As an instructive example, let us consider SU(2) theory with 4 flavors, whose perturbative symmetry is SO(8)×U(1).In Figure 8, three different unHiggsed 5-brane configurations are considered and each gives rise to different manifest global symmetries as depicted.The red lines in Figure 8 are D5-branes which we perform the defect Higgsing.The 5-brane configurations (a) and (b) are, in particular, instructive.Both unHiggsed 5-brane webs are the same as SU(3) 0 gauge theory with 6 flavors, their resulting global symmetries however are different.Global symmetry for (a) is SU(4)×SU(2)×SU(2), while that for (b) is SU(4)×U(1)×SU(2).Here SU(4) is the symmetry coming from the interchange of four flavor masses, and SU(2) arises from two parallel (2, 1) 5-branes in Figure 8 (a) and (b), which is associated with instanton.Now the difference can be explained as follows.In  4)×SU( 2)×SU( 2), (b) SU( 4)×U( 1)×SU(2), and (c) SU( 5)×U (1).The red lines are D5-branes where the defect Higgsing is applied.A white dot is 7-brane.
(a), the defect Higgsing is applied to internal D5-brane, which leaves two parallel (0,1) 5-branes on the bottom symmetric, accounting for the SU(2) part of its global symmetry.
In (b), on the other hand, the defect Higgsings is applied to an outer D5-brane which makes two parallel (0,1) 5-branes on the bottom distinguishable, hence breaking SU(2) into U(1).The unHiggsed 5-brane configuration (c) is also noticeable.The resulting global symmetry is SU( 5)×U(1), which is obtained from an SU(4) − 1 2 gauge theory with 7 flavors, with usual Higgsing in the upper part and also with a defect Higgsing in the lower part.From these three examples, it is clear that global symmetry of defect Higgsings comes from global symmetry structure of unHiggsed 5-brane web, which could be simply perturbative symmetry or one of the maximal compact subgroup of the enhanced global symmetry, SO (10) in this particular example.
As there could be many more unHiggsed 5-brane configurations that one can geometrically engineer, we expect different global symmetry depending on unHiggsed 5-brane that we used for the computations.

Conclusion
In this paper, we computed defect partition function of E-string theory on a circle, from 5brane webs by applying the defect Higgsing, based on two 5-brane configurations: one with two O5-planes and the other without O5-planes.Though two 5-brane configurations look different, both parameter phases actually describe 5d SU(3) 0 gauge theory with 10 flavors, as the 5-brane web with two O5-planes is deformed to describe SU(3) gauge theory phases.We however referred to the one with O5-planes as Sp(2) gauge theory with 10 flavors just to distinguish natural 5-brane web, without orientifolds, for SU(3) gauge theory with 10 flavors.As 6d E-string theory on a circle is realized as 5d Sp(1) gauge theory with 8 flavors, we applied a suitable defect Higgsing on these 5-brane webs to obtain the surface defect partition function for 5d Sp(1) gauge theory with 8 flavors.The resulting partition function has the defect modulus which corresponds to the position on 5-brane where the defect is inserted.With the defect modulus, the partition function can be understood as the open topological string partition function.We compared our defect partition function with 6d elliptic genus result where the same defect Higgsing is implemented.We carried out our computation of the defect partition function up to 2-instantons and confirmed that the the results agree up to that order.
For the 5-brane configurations that we considered in the paper, an insertion of the defect breaks global symmetry from E 8 to SO (16).This can be explicitly seen from the defect partition function obtained from the elliptic genus where the theta function at a given instanton order is invariant under the SO (16) Weyl transformation rather than the E 8 transformation.From the partition function of 5d Sp(1) gauge theory with 8 flavors, the SO( 16) corresponds to the perturbative SO (16) flavor symmetry.The corresponding 5-brane web suggests that flavors are not affected by the defect and hence 8 flavor branes can be put closer to one of O5-planes so that they enjoy an SO( 16) symmetry.In this way, one can see that regardless of the number of the defects, SO( 16) global symmetry remains.Together with U(1) coming from the KK modes, the defect partition function hence has SO(16) × U(1) KK symmetry.
We note that there are different ways of realizing 5d brane system with defects, as discussed in section 4. Depending how we implement the defect Higgsing, we may see different global symmetry structure.It is however that such apparent global symmetries are in fact a subgroup of the enhanced global symmetry, as one can readily restore the enhanced global symmetry when setting the number of defects to zero.Even for the case of the E-string theory with defects, global symmetry may restore to be E 8 if there is a nontrivial unHiggsed 5-brane configuration or nontrivial reparametrization involving the defect modulus such that it respects E 8 Weyl transformation.It is hence interesting to further study how global symmetry structure is broken or hidden in the presence of defects.For instance, quantization of SW curve or associated integrable system [9,[40][41][42].
Some defects an be taken away by taking the defect insertion to infinity.In our setup, however, as we inserted the defect in between two NS5 branes, it is not possible to decouple the defects.Instead, one can set the number of the defects to zero, then one naturally recovers the E-string partition on a circle which restores E 8 global symmetry.
It is straightforward to generalize our computation to higher rank gauge theories.For instance, 6d N = 1 Sp(N ) gauge theory with 2N + 8 flavors with a tensor compactified on a circle can be realized as a 5-brane web with two O5-planes describing 5d N = 1 Sp(N +1) gauge theory with 2N + 8 flavors.As we can apply the defect Higgsing to it, we expect to obtain 5d Sp(N ) gauge theory with with 2N + 6 flavors and defects.Global symmetry for the theory with defects would be an SO(4N + 12) × U(1) KK , as the corresponding 5-brane configuration suggests that there are O5-planes with 2N + 6 flavor D7-branes, and the instanton gives a U (1).
We can also consider the defect Higgsing for the system with less flavors.Starting from a 5d KK theory with defects, say 5d N = 1 Sp(N + 1) gauge theory with 2N + 8 flavors and defects, one can decouple flavors one by one to get 5d N = 1 Sp(N + 1) gauge theory with N f < 2N + 8 flavors and defects.For example, we worked out explicitly the partition function for 5d Sp(2) gauge theory with 9 and 8 flavors in Appendix D.
The efficient way to calculate the partition function is to use the operator formalism explained in [43].After some computations, one finds obtain (2.12).To reach the result, we use the analytic continuation formula,

D Flavor decoupling limit
To consider what global symmetry is preserved in the presence of the defect in the 5d gauge theories with N f < 10 flavors, we shall take the decoupling limit and consider the defect with   (1 , (E.7a) Then we find , (E.8) under following relation, The corresponding web diagrams are as follows.
. The web diagrams after the geometric transition.Since we move two D7-branes, the shape of framing after moving the D7-branes is more complicated than previous case depicted in Figure 20.

Figure 1 .
Figure 1.(a) The Tao diagram corresponding to the 5d N = 1 SU(3) gauge theory with 10 flavors whose spiral structure continues infinitely.(b) Its building block.

Q 1 <
l a t e x i t s h a 1 _ b a s e 6 4 = " X C K p A l 4X P H k X x R H R / 0 F M w w z 1 J X k = " > A A A C h H i c h V G 7 S g N B F D 2 u G m N 8 J G o j 2 A R D J I W E W Y 0 o F h K w s T T G P C C G s L u O c c m + 2 N 0 E Y s g P C L a m s F K w E H t / w M Y f s P A T x D K C j Y U3 m w X R Y L z D z J w 5 c 8 + d M 1 z Z 0 l T H Z e x l R B g d G w 9 M B C d D U 9 M z s + H I 3 r F 2 n 9 I / M B g a C g 9 H R k b H x q O x i c l d z 6 6 7 B s 8 b t m m 7 + 7 r m c V N Y P C + F N P m + 4 3 K t p p t 8 T 6 9 u d O 7 3 7 B 1 m 5 p l n 3 u e d Z 3 g 1 2 x C u Z O y l T + k f G B w a D o y M j o 0 H J 0 L h y a m c a 9 U c n W d 1 y 7 C c P U 1 1 u S F M n p V C G n z P d r h a 1 Q y + q 1 X W 2 v e 7 d e 6 4 w j J 3 5 L y k o n V e n z N d P 4 E c r 6 F 3 B f k x k = < / l a t e x i t > Q1 < l a t e x i t s h a 1 _ b a s e 6 4 = " T x s L F E L + h R 5 p R e d j L f F + 5 3 / P W 0 4 = " > A A A C i 3 i c h V G 7 S g N B F D 2 u 7 0 c 0 a i P Y B E M k V Z j V g C I W g g i W e R g N R g m 7 m 1 E H N 7 v L 7 i S g I T 9 g L V i I g o K F 2 P s D N v 6 A h Z 8 g l g o 2 F t 7 d L I g G 4 x 1 m 5 s y Z e + 6 c 4 e q O K T z J 2 H O X 0 t 3 T 2 9 c / M D g 0 P B I Z H Y u O T 2 x 6 d s 0 1 e M G w T d s t 6 p r H T W H

3 <
f w I Z e 0 L b k G T G A = = < / l a t e x i t > µ l a t e x i t s h a 1 _ b a s e 6 4 = " d p r U n v e 2 p S 8 P S o m N e i y O l m G L d r 4 D 9 j 4 A x Z + g l h G s L H w 7 m Z B V I x 3 m J k z Z + 6 5 c 4 Z r e J Y I J G P P C a W n t 6 9 / I D k 4 N D w y O p Z K j 0 9 s B m 7 d N 3 n Z d C 3 X 3 z b 0 g F v C 4 W U p p M W 3 P Z / r t m H x L e N w J b z f a n A / E K 6 z I Y 8 8 v m f r N U d U h a l L o s q 7 d r 2 i V d J Z p r I o M r + B F o M s 4 i i 6 6 X v s Y h 8 u T N R h g 8 O B J G x B R 0 B j B x o Y P O L 2 0 C T O J y S i e 4 4 W h k h b p y x O G T q x h 7 T W 6 L Q T s w 6 d w 5 p B p D b p F Y u m T 8 o M Z t k T u 2 V t 9 s j u 2 A v 7 + L N W M 6 o R e j m i 3 e h o u V d J n U y t v / + r s m m X O P h S d f U s U c V S 5 F W Q d y 9 i w l + Y H X 3 j + K y 9 v r w 2 2 5 x j 1 + y r O p 5 a p V c 0 m j Y p 1 x B l L + y e 9 d g z e 2 C v 7 P P P W m 2 v R t 9 L i 3 Z l o O V W L X S 2 s v f x r 0 q n 3 c X x t 2 q k Z x d 1 b H l e B X m 3 P K b / C 3 W g b 5 5 e 9 P a 2 8 9 H 2 B r t l b + T / h n X Z E / 3 A a L 6 r d z m e v x 7 h R y E v H W p P 4 n c z h s F B M p 5 I x V O 5 V C Q T 8 x s V w C r W E a N u p J H B L r I o U H W B c 1 z i S g p I c W l T S g 9 S p T F f s 4 w o B F P k E s I 9 h Y e H e z I C r q H W b m z J l 7 7 p z h a o 4 h P J + x b k w a G B w a H o m P J s b G J y a T q a n p m m c 3 X J 0 r u m 3 Y 7 p a m e t w Q F l d 8 4 R t 8 y 3 G 5 a m o G 3 9 S O 1 o L 7 z S Z 3 P W F b G 3 7 L 4 b u m e m C J f a G r P l H K j t W o 5 + q p D M u y M N I / g R y B D K I o 2 a l 7 7 G A P N n Q 0 Y I L D g k / Y g A q P x j Z k M D j E 7 a J N n E t I h P c c H S R I 2 6 A s T h k q s U e 0 H t B p O 2 I t O g c 1 v V C t 0 y s G T Z e U a S y w J 3 b L e u y R 3 b F n 9 v 5 r r X Z Y I / D S o l 3 r a 7 l T T 5 7 M V t / + V Z m 0 + z j 8 V P 3 p 2 c c + V k K v g r w 7 I R P 8 Q u / r m 8 d n v e p q Z a G 9 y K 7 Z C / m / Y l 3 2 Q D + w m q / 6 T Z l X L v / w o 5 G X D r V H / t 6 M n 6 C W y 8 r 5 b L 6 c z x S X o k b Y D C I q 6 B D n E l I c e 4 5 u v C T t k l Z n D J E Y u u 0 H t O p 5 L I a n Q c 1 L U c t 0 y s q T Z O U I U T Y C 7 t n f f b M H t g r + / y z V s e p M f D S p l 0 a a r l R D Z y t Z D / + V T V o t 3 H y r R r r 2 U Y N O 4 5 X h b w b D j P 4 h T z U t 0 6 v + t n d T K S z z m 7 Z G / m / Y T 3 2 R D / Q W u / y X Z p n r s f 4 k c h L l 9 o T + 9 2 M U Z C P R 2 O J a C K d C C e 3 3 E b 5 s I o 1 b F A 3 t p H E A V L I U f U a z n G B S 8 E r b A p x I T F M F T y u Z h k / Q t j 7 A p r M k C g = < / l a t e x i t > ø < l a t e x i t s h a 1 _ b a s e 6 4 = " N / H p f Y + 1 U 7 6 c r z K V r n G y 6 U e R Z c I Y D C I q 6 B D n E l I c e 4 5 u v C T t k l Z n D J E Y u u 0 H t O p 5 L I a n Q c 1 L U c t 0 y s q T Z O U I U T Y C 7 t n f f b M H t g r + / y z V s e p M f D S p l 0 a a r l R D Z y t Z D / + V T V o t 3 H y r R r r 2 U Y N O 4 5 X h b w b D j P 4 h T z U t 0 6 v + t n d T K S z z m 7 Z G / m / Y T 3 2 R D / Q W u / y X Z p n r s f 4 k c h L l 9 o T + 9 2 M U Z C P R 2 O J a C K d C C e 3 3 E b 5 s I o 1 b F A 3 t p H E A V L I U f U a z n G B S 8 E r b A p x I T F M F T y u Z h k / Q t j 7 A p r M k C g = < / l a t e x i t > ø < l a t e x i t s h a 1 _ b a s e 6 4 = " N / H p f Y + 1 U 7 6 c r z K V r n G y 6 U e R Z c I 5 c 4 a r u 6 b w J W P P M a W n t 6 9 / I D 4 4 N D w y O p Z I j k 9 s + U 7 g G b x k O K b j b e u a z 0 1 h 8 5 I U 0 u T b r s c 1 S z d 5 W a + v t u / L D e 7 5 w r E 3 5 a H L q 5 a 2 b 4 s 9 r t D o 6 + V H 9 6 d l D B s u d V I e + m x 7 i / k D v 6 2 s l 5 O 7 u 6 N d u Y Y z f s l f x f s x Z 7 p B / o t T f 5 N s O 3 L v 7 w I 5 E X t z 2 J n 8 3 o B v m F e C I Z T 2 a S 0 V T M b 1 Q Q U 5 h B j L q x h B Q 2 k U a O q m s 4 w y W u h L C w I K w I a 5 1 U I e B r J v A t h I 1 P 0 j W S Y A = = < / l a t e x i t > Q (4) 7 < l a t e x i t s h a 1 _ b a s e 6 4 = " / M c E A w J A U 8 6 M v w T q X D c 4 4 L U T 8 L o 1 C N b t r y s Z + Z X f a p D b 9 A 6 4 O D j i Q O I i 7 P + D i D z j 4 C e J Y i Y u D d 7 e b C I J 3 M j P P P P M + 7 z y T e X T Y w S 8 W 5 2 5 8 y Z e + 6 e y Z F N l d m c k L F P W F h c W l 7 x r w a C a + s b

Figure 3 .
Figure 3. (a): A Type IIA brane configuration for 6d Sp(1) gauge theory with 10 flavors and a tensor.(b): A Type IIB brane configuration which is T-dual of (a).
1 G 7 I m r / 2 a b q D W + B S D u 8 P K J G b o i e 6 o T Y 9 0 T y 1 6 / 7 V W I 6 j h e 6 n z r H a 0 w i 7 G z y b y b

3 <
r d U I a v h e 6 j y r b a 2 w y 7 G z x P b 7v 6 o K z x 6 O v l R / e v Z w g J X A q 8 7 e 7 Y D x b 6 G 1 9 b W T i 9 b 2 2 t Z M Y 5 Z u 6 J X 9 X 9 M L P f I N z N q b d p s V W 1 e I 8 g f I P 5 + 7 E + Q X 0 v J S e j W 7 l F p P h l / R j 0 l M Y 5 7 f e x n r 2 E Q G O T 7 3 C O e 4 w G W k K c W k C S n R T p U i o W Y c 3 0 K a + g T W k o q 6 < / l a t e x i t >x 0 l a t e x i t s h a 1 _ b a s e 6 4 = " G w v e G N l v T J e L 4 A P z S A P K A h e W w c 8

2 <
r d U I a v h e 6 j y r b a 2 w y 7 G z x P b 7v 6 o K z x 6 O v l R / e v Z w g J X A q 8 7 e 7 Y D x b 6 G 1 9 b W T i 9 b 2 2 t Z M Y 5 Z u 6 J X 9 X 9 M L P f I N z N q b d p s V W 1 e I 8 g f I P 5 + 7 E + Q X 0 v J i e j W 7 m F p P h l / R j 0 l M Y 5 7 f e x n r 2 E Q G O T 7 3 C O e 4 w G W k K c W k C S n R T p U i o W Y c 3 0 K a + g T a l I q 8 < / l a t e x i t >x 0 l a t e x i t s h a 1 _ b a s e 6 4 = " C r D u k M k 5 4 g i y u e l / 3 2 b l G 2

2 < l a t e x i t s h a 1 _ b a s e 6 4 =
n 3 b 7 2 d 1 M t L P G b t g r + b 9 m L + y R b q C 1 3 + T b N M 9 c w U 8 f E P / 5 3 M M g n 4 j F N 2 M 7 6 c 1 I M u x 9 x S S W s Y o N e u 8 t J L G P F H J 0 b h 0 X 6 O L S 1 x M C w p I Q G q Q K P k + z i G 8 h r H w C 0 o 6 K u A = = < / l a t e x i t >x 0 " x 4 K l m I h a y u J N k 7

1 <
G S P b A b 1 m T 3 7 J Y 9 s f d f a 9 W 9 G i 0 v N Z r l t p Z b x f D R W O b t X 5 V O s 4 u 9 T 9 W f n l 2 U s O B 5 V c m 7 5 T G t W y h t f f X w p J l Z S k / W p 9 g l e y b / F + y R 3 d E N j O q r c p X i 6 V O E 6 A P E 7 8 / 9 E 2 z M J c R k Y j G V j C / H / K 8 I I o o J z N B 7 z 2 M Z q 1 h D l s 7 V c Y w z n A d e h F E h K o y 3 U 4 W A r x n B l x C m P g D / Q I y e < / l a t e x i t >x 0 l a t e x i t s h a 1 _ b a s e 6 4 = " v 6 Y T i 1 6 C e N e Z A a M U h q 9

l a t e x i t > y 0 10 < l a t e x i t s h a 1 _ 6 < l a t e x i t s h a 1 _
b a s e 6 4 = " L n t S o a g o H c z q + g R f 3U L c M Q V 6 t Z A = " > A A A C b 3 i c h V H L S s N A F D 2 N r 1 p f V R c K g h S L r 4 X l R g Q f q 4 I b l 7 7 6 A B 8 l i V M N p k l I 0 k I N / Q E / Q B c u f I C I + B l u / A E X f o K 4 E g U 3 L r x N A 6 K i 3 m F m z p y 5 5 8 6 Z G d U 2 d N c j e o h I T c 0 t r W 3 R 9 l h H Z 1 d 3 T 7 y 3 L + t a Z U c T G c 0 y L C e v K q 4 w d F N k P N 0 z R N 5 2 h F J S D Z F T 9 x f r + 7 m K c F z d M t e 9 q i 2 2 S s q u q R d 1 T f G Y 2 q w W f J l q 4 9 v + l F w r x J O U o i A S P 4 E c g i T C W L b i V 9 j E D i x o K K M E A R M e Y w M K X G 4 b k E G w m d u C z 5 z D S A / 2 B W q I s b b M W Y I z F G b 3 e d z l 1 U b I m r y u 1 3 Q D t c a n G N w d V i Y w S v d 0 T S 9 0 R z f 0 S O + / 1 v K D G n U v V Z 7 V h l b Y h Z 7 D w b W 3 f 1 Ul n j 3 s f a r + 9 O y h i L n A q 8 7 e 7 Y C p 3 0 J r 6 C s H x y 9 r C 6 u j / h h d 0 B P 7 P 6 c H u u U b m J V X 7 X J F r J 4 g x h 8 g f 3 / u n y A 7 n Z J n U v M r M 8 l 0 I v y K K I Y w g g l + 7 1 m k s Y R l Z P h c G 0 c 4 x V n k W R q Q h q U w V 4 q E m n 5 8 C W n y A 8 t A j e Q = < / l a t e x i t > y 0 b a s e 6 4 = "

8 < l a t e x i t s h a 1 _
D W I M E m 7 4 8 R D C Y x O 3 B J c 4 i p H r 7 H A 0 E S V u l L E 4 Z E r F l + p d o l f d Z n d b t m r a n V u g U j Y Z F y g j m 2 C O 7 Z S 3 2 w O 7 Y E 3 v / t Z b r 1 W h 7 q d M s d 7 T c L I S O J 9 N v / 6 o q N D s 4 / F T 9 6 d l B E a u e V 5 W 8 m x 7 T v o X S 0 d e O T l v p 9 d S c O 8 + u 2 D P 5 v 2 R N d k 8 3 0 G u v y n W S p 8 4 Q p A c Q v 7 f 7 J 8 g u x c X l + F p y O b o R 8 Z + i H 1 O Y R Y z 6 v Y I N b C G B j N e x E 5 z j I v A i T A j T w k w n V Q j 4 m n F 8 C S H 2 A U 8 i j b A = < / l a t e x i t > y 0 b a s e 6 4 = " I V 1 f t 1 o i J 4 Z o + 6 w 7 N C n Q m e J n t y I = " > A A A C b

9 <
8 T P c + A M u + g n i R q j g x o W 3 a U C 0 q D d M 5 s y Z e + 6 c u S O b m m o 7 j D U C Q l d 3 T 2 9 f s D 8 0 M D g 0 H I 6 M j G 7 b R s V S e F Y x N M P a l S W b a 6 r O s 4 7q a H z X t L h U l j W + I 5 f W W v s 7 V W 7 Z q q F v O T W T 7 5 e l o q 4 W V E V y i M r V 8 u 5 S f e 7 A n R f r + U i M J Z k X 0 U 4 g + i A G P 9 J G 5 B Z 7 O I Q B B R W U w a H D I a x B g k 1 f D i I Y T O L 2 4 R J n E V K 9 f Y 4 6 Q q S t U B a n D I n Y E v 2 L t Mr 5 r E 7 r V k 3 b U y t 0 i k b D I m U U c f b E 7 l i T P b J 7 9 s w + f q 3 l e j V a X m o 0 y 2 0 t N / P h k 4 n N 9 3 9 V Z Z o d H H 2 p / v T s o I A l z 6 t K 3 k 2 P a d 1 C a e u r x 2 f N z Z W N u D v L r t k L + b 9 i D f Z A N 9 C r b 8 p N h m + c I 0 Q P I P 5 s d y f Y X k i K q e R y J h V b j f p P E c Q k Z p C g f i 9 i F e t I I + t 1 7 B Q X u A y 8 C u P C l D D d T h U C v m Y M 3 0 J I f A J R K Y 2 x < / l a t e x i t > y 0 l a t e x i t s h a 1 _ b a s e 6 4 = " Q x m S U 5 P B b X T A n / c f a p z r S + Y Q g 5 s y Z e + 6 c u S O b m m o 7 j D U C Q l d 3 T 2 9 f s D 8 0 M D g 0 H I 6 M j G 7 b R s V S e F Y x N M P a l S W b a 6 r O s 4 7q a H z X t L h U l j W + I 5 f W W v s 7 V W 7 Z q q F v O T W T 7 5 e l o q 4 W V E V y i M r V 8 u 5 y f e 7 A n R f r + U i M J Z k X 0 U 4 g + i A G P 9 J G 5 B Z 7 O I Q B B R W U w a H D I a x B g k 1 f D i I Y T O L 2 4 R J n E V K 9 f Y 4 6 Q q S t U B a n D I n Y E v 2 L t Mr 5 r E 7 r V k 3 b U y t 0 i k b D I m U U c f b E 7 l i T P b J 7 9 s w + f q 3 l e j V a X m o 0 y 2 0 t N / P h k 4 n N 9 3 9 V Z Z o d H H 2 p / v T s o I A l z 6 t K 3 k 2 P a d 1 C a e u r x 2 f N z Z W N u D v L r t k L + b 9 i D f Z A N 9 C r b 8 p N h m + c I 0 Q P I P 5 s d y f Y X k i K q e R y J h V b j f p P E c Q k Z p C g f i 9 i F e t I I + t 1 7 B Q X u A y 8 C u P C l D D d T h U C v m Y M 3 0 J I f A J T M I 2 y < / l a t e x i t > (a) j j Z 9 j 4 A x Z 8 g r F T E x s L L 8 s m R o 1 6 J z N z 5 s w 9 d 8 7 M K L a u u Y K x T k D q 6 x 8 Y H B o e C Y 6 O j U + E w p N T + 6 7 V c F S e V S 3 d c v K K 7 H J d M 3 l W a E L n e d v h s q Ho P K f U t r v 7 u S Z 3 X M 0 y 9 8 S R z U u G X D G 1 s q b K g q h S 0 Z B F t e z I t V b 9 Z O k g H G N x 5 k X 0 J 0 j 4 I A Y / U l b 4 G k U c w o K K B g x w m B C E d c h w q R W Q A I N N X A k t 4 h x C m r f P c Y I g a R u U x S l D J r Z G Y 4 V W B Z8 1 a d 2 t 6 X p q l U 7 R q T u k j G K B P b A b 9 s L u 2 S 1 7 Z O + / 1 m p 5 N b p e j m h W e l p u H 4 R O I 7 t v / 6 o M m g W q n 6 o / P Q u U s e 5 5 1 c i 7 7 T H d W 6 g 9 f f P 4 7 G V 3 M 7 P Q W m S X 7 I n 8 t 1 m H 3 d E N z O a r e p X m m X M E 6 Q M S 3 5 / 7 J 9 h f j S e S 8 Y 1 0 M r Y V 9 b 9 i G D O Y x z K 9 9 x q 2 s I M U s n R u H W e 4 Q D v w L E W k W W m u l y o F f M 0 0 v o S 0 8 g F V n o 8 c < / l a t e x i t > q 0

Figure 4 .
Figure 4.A 5-brane web diagram description of 5d N = 1 Sp(2) gauge theory with 10 flavors.The dashed lines and the solid lines denote the O5-planes and (p, q) 5-branes, respectively.The variables x I (I = 1, 2, 3) and y i (i = 1, 2, ..., 10) denote the position of the horizontal legs measured from the middle of the dashed line, and q denotes the instanton factor.

Figure 5 .
Figure 5. Higgsing in a Tao diagram for SU(3) theory with 10 flavors.Since the framing denoted by the red lines decouples from the remaining web diagram, the part of the summation of the Young diagram is decoupled from other summations.

Figure 6 .
Figure 6.A web diagram description of a usual Higgsing.By setting the Kähler parameters to one corresponding to equate the Coulomb branch moduli and the fundamental masses in order to enter the Higgs branch, we can remove the brane depicted by the red line, and the resulting web diagram gives 5d N = 1 Sp(1) gauge theory with 8 flavors.
t b L e u S / W h e P q l r n l N W y x U 1 U O T H 1 f 1 x S P q a y 8 G 4 1 T g v y I / Q R y A O I I I m N F 7 7 G N P V j Q U E M V A i Y 8 x g Y U u D z K k E G w m d t B k z m H k e 7 f C x w j z N o a Z w n O U J i t 8 H r A p 3 L A m n z u 1 H R 9 t c a v G D w d V s Y w S 0 9 0 S 2 1 6 p D t 6 p v d f a z X 9 G h 0 v D d 7 V r l b Y u 5 G T q d z b v 6 o q 7 x 4 O P 1 V / e v a w j 2 X f q 8 7 e b Z / p / E L r 6 u t H F + 3 c y u Z s c 4 6 u 6 k z 9 9 w 5 w 9 U d U 3 i S s c c O p b O r u 6 e 3 r z 8 y M D g 0 H I 2 N j B Y 8 u + 4 a P G / Y p u 0 W d c 3 j p r B 4 X g p p 8 q L j c q 2 m m 3 x b r 6 7 5 9 9 s N 7 n r C t r b k o c N 3 a 1 r F E m V h a J K o 3 I w 2 u x d L s C Q L I v 4 T q C F I I I w N O 3 a L H e z D h o E 6 a u C w I A m b 0 O D R K E E F g 0 P c L p r E u Y R E c M 9 x j A h p 6 5 T F K U M j t k p r h U 6 l k L X o 7 N f 0 A r V B r 5 g 0 X V L G M c U e 2 D V 7 Y f f s h j 2 x 9 1 9 r N Y M a v p d D 2 v W 2 l j t 7 0 Z P x 3 N u / q h r t E g e f q j 8 9 S 5 S x F H g V 5 N 0 J G P 8 X R l v f O G q 9 5 J a z U 8 1 p d s m e y f 8 F e 2 R 3 9 A O r 8 W p c b f L s 6 2 s l F O 7 O e n m n M s m v 2 Q v 6 v W I s 9 0 A / M 2 q t + k + L p y z / 8 a O S l S e 1 J f G / G T 5 B b j C e W 4 o u p 5 d j m l t + o I K Y w j T n q x i o 2 s Y M k s l S 9 h D O c 4 0 L p V x a U J W W l m 6 o E f M 0 E v o S y 8 Q G T e J A t < / l a t e x i t > q M

Figure 7 .
Figure 7.The 5-branes with the defect introduced by defect Higgsing.The blue dashed lines denote the defects.

Figure 21 .
Figure 21.The web diagrams which reduce to SU(2) gauge theory with θ = 4π and θ = 0 in the presence of the defects after the geometric transition.
t e x i t s h a 1 _ b a s e 6 4 = " Q R L / r i h 7 S N D E 1 H t 9 Q a b 3 t J W p O A Q = " > A A A C Z n i c h V F N S w J B G H 7 c v s x K r Y i C L p I Y d p E x h D 5 O Q p e O f u Q Hm M j u N t r i u r v s r o J J f y D o m o d O B R H R z + j S H + j g P y g 6 G n T p 0 O u 6 E C X V O 8 z M M 8 + 8 z z v P z E i G q l g 2 Y z 2 P M D Y + M T n l n f b N z M 7 5 A 8 H 5 h b y l N 0 2 Z 5 2 R d 1 c 2 i J F p c V T S e s x V b 5 U X D 5 G J D U n l B q u 8 N 9 g s t b l q K r h 3 Y b Y O X G 2 J N U 6 q K L N p E Z a P S R i U Y Z j H m R G g U x F 0 Q h h s p P X i L Q x x B h 4 w m G u D Q Y B N W I c K i V k I c D A Z x Z X S I M w k p z j 7 H K X y k b V I W p w y R 2 D q N N V q V X F a j 9 a C m 5 a h l O k W l b p I y h A h 7 Y n e s z x 7 Z P X t h H 7 / W 6 j g 1 B l 7 a N E t D L T c q g b O V 7 P u / q g b N N o 6 / V H 9 6 t l H F t u N V I e + G w w x u I Q / 1 r Z N u P 7 u b i X T W 2 T V 7 J f 9 X r M c e 6 A Z a 6 0 2 + S f P M J X z 0 A f G f z z 0 K 8 p u x e C K 2 k 0 6 E k 1 H 3 K 7 x Y x R q i 9 N 5 b S G I f K e T o 3 B r O c Y G u 5 1 n w C 0 v C 8 j B V 8 L i a R X w L I f Q J s 9 2 K O Q = = < / l a t e x i t > (a)< l a t e x i t s h a 1 _ b a s e 6 4 = " c G n C 9 0 x i e p u q R t G y L 5 i B r n g z w b 4= " > A A A C Z n i c h V F N S w J B G H 7 c v s x K r Y i C L p I Y d p E x h D 5 O Q p e O f u Q H m M j u N t r i u r v s r o J J f y D o m o d O B R H R z + j S H + j g P y g 6 G n T p 0 O u 6 E C X V O 8 z M M 8 + 8 z z v P z E i G q l g 2 Y z 2 P M D Y + M T n l n f b N z M 7 5 A 8 H 5 h b y l N 0 2 Z 5 2 R d 1 c 2 i J F p c V T S e s x V b 5 U X D 5 G J D U n l B q u 8 N 9 g s t b l q K r h 3 Y b Y O X G 2 J N U 6 q K L N p E Z a P i R i U Y Z j H m R G g U x F 0 Q h h s p P X i L Q x x B h 4 w m G u D Q Y B N W I c K i V k I c D A Z x Z X S I M w k p z j 7 H K X y k b V I W p w y R 2 D q N N V q V X Fa j 9 a C m 5 a h l O k W l b p I y h A h 7 Y n e s z x 7 Z P X t h H 7 / W 6 j g 1 B l 7 a N E t D L T c q g b O V 7 P u / q g b N N o 6 / V H 9 6 t l H F t u N V I e + G w w x u I Q / 1 r Z N u P 7 u b i X T W 2 T V 7 J f 9 X r M c e 6 A Z a 6 0 2 + S f P M J X z 0 A f G f z z 0 K 8 p u x e C K 2 k 0 6 E k 1 H 3 K 7 x Y x R q i 9 N 5 b S G I f K e T o 3 B r O c Y G u 5 1 n w C 0 v C 8 j B V 8 L i a R X w L I f Q J s d y K O A = = < / l a t e x i t >